What is the intuition behind the TD(0) equation with average reward, and how is it derived?
This is simply from definition of return in average reward setting (look at equation $10.9$). The "standard" TD error is defined as
\begin{equation}
TD_{\text{error}} = R_{t+1} + V(S_{t+1}) - V(S_t)
\end{equation}
In average reward setting, average reward $r(\pi)$ is subtracted from reward at $t$, $R_t$, so TD error in this case is
\begin{equation}
TD_{\text{error}} = R_{t+1} - \bar R_{t+1} + V(S_{t+1}) - V(S_t)
\end{equation}
where $\bar R_{t+1}$ is estimate of $r(\pi)$.  

You can use $\delta_t$ in all 3 updates because neither of these updates depend on each other. For example if you update $\mathbf w$ you don't use that then to update $\mathbf \theta$, or if you update $\bar R$ you don't use updated version to update $\mathbf w$ or $\mathbf \theta$ so you're not introducing additional bias. In each separate update you also don't have $\delta_t$ present multiple times so that you require multiple sampling per timestep to get the unbiased update.  

Additionally this is semi-gradient algorithm, it uses bootstrapped estimate $V_{t+1}$ but it doesn't calculate full derivative with respect to it aswell, only with respect to $V_t$ so the algorithm is biased by default, but works well enough in practice for linear case.